The existence of the magnetic-field-induced liquid-to-solid phase transition in an extreme quantumlimit 2D electron plasma is established for electrons at a high-quality GaAs/GaAlAs heterojunction by detection of a gapless magnetophonon excitation branch with radio-frequency spectroscopy. The phase diagram, determined for Wigner-Seitz to Bohr radius ratio 1.6 <») limit. PACS numbers: 73.40.Kp, 64.70.-p, 73.20.MfA zero-temperature electron plasma confined to a plane shows unusually varied physical properties as a function of density ris and perpendicular magnetic field H, In zero field and high density, the zero-point energy arising from the Pauli exclusion principle dominates, resulting in a liquid state with Fermi energy Ef^^h^/ma^, which corresponds to the energy of confinement in the Wigner-Seitz radius, a, defined by Ka^^X/ris. As the density is reduced the Coulomb interaction Vc^e^lea in the medium of dielectric constant e gains in importance, ultimately ordering the electrons into a crystal. Wigner * predicted this quantum phase transition long ago, but it has never been seen for want of a suitable physical system.In a magnetic field, however, each single-particle state is already confined by the Lorentz force to an area --(po/H'^lKli occupied by a flux quantum ^o^hc/e Un'^ihc/eH)^^^ is the zero-point cyclotron radius] and takes on discrete energy values (n+ j)hcoc where h(Oc -h^/ml^ -heH/mc is the cyclotron energy. Each of these Landau cyclotron oscillators has an average share of v=2(////a)^ electrons. Increasing the field decreases this filling factor v, successive Landau levels empty, and, in the presence of moderate disorder, one observes the integrally quantized Hall resistance.^ When V < 1 there is spatial freedom for all the electrons for the same zero-point energy. This greatly favors Coulomb-induced correlation. The many-body ground state is a correlated liquid giving rise to the fractional quantum Hall effect^ until, below a critical filling factor estimated between y and j,^ a new ground state with long-range crystallike correlation is expected. To date, this transition, too, has eluded unambiguous observation.The present experiment gives positive evidence of this magnetically induced Wigner solid. Advances in the technology of modulation-dopedGaAs/GaAlAs heterojunctions have made available very high-quality 2D quantum electron systems. Some time ago,^ it was already suggested that the abnormal behavior observed in the dc transport for v < j could be a result of a magnetically induced Wigner solid. But to be affirmative required a much more specific type of experiment. Our experiment tests for a defining property of a solid-its rigidity to shear. An unbounded charged 2D liquid has only compressional (plasmon) excitations in zero field whose frequency-wave-vector relation is (ol^lKe^risqlem. A perpendicular magnetic field i...

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The 2D quantum system of electrons at a GaAs/GaAlAs heterojunction in high magnetic field at low temperature is shown to exhibit conduction typical of pinned charge-density waves. Crossover from Ohmic conduction occurs on the same boundary at which radio-frequency resonances signal the onset of transverse elasticity. A further small non-Ohmic region is isolated from the main area by a v-j quantum-Hall-effect phase. The relationship found between the threshold conduction field and the resonance frequency is well accounted for by a model of a pinned electron crystal.

We have shown that the collective modes of a bounded two-dimensional plasma in a magnetic field exhibit unexpected features which are dynamical manifestations of the Hall effect. In high fields we observed the appearance of a novel magnetically localized one-dimensional wave which propagates along the perimeter of the plasma. PACS numbers: 73.20.Cw, 71.45.Gm, 72.15.Gd The Lorentz force acting on a bounded system of charge carriers in a magnetic field can induce charge accumulation at the boundaries. The Hall resistance, which is a stationary manifestitation of this phenomenon, has traditionally been exploited to measure the charge carrier density. But it has not yet been recognized that the dynamics of a plasma in a magnetic field in the presence of density inhomogeneity (in particular boundaries) has physical aspects which are more general than the Hall resistance. In this Letter we present the first theoretical and experimental results on the dynamical Hall effect in a twodimensional classical plasma. The theoretical results obtained in a linearized frictionless hydrodynamic approach will be compared with measurements of the plasmon mode frequencies of electrons confined on a liquid helium surface in a cylindrical cell placed in a normal magnetic field Hz. The spectrum and its behavior with magnetic field can be seen in Figs. 1 and 2. An unusual feature is the existence of modes whose frequencies decrease with field, in contradiction to the generally used infinite-geometry rule (o 2 = a)p(H = 0) +w c 2 where w p (/f = 0) is the zerofield plasmon frequency and co c = eH/mc is the cyclotron frequency. 1 We shall show that this decrease is one of the dynamical manifestations of the Hall effect.We suppose the electrons of mass m and charge e to be confined to a single region of the plane z = 0 by a set of externally imposed potentials which determine the equilibrium charge distribution n s (r) = n 0 o-(r) where r is a two-dimensional position vector. We will now show that the hydrodynamic problem can be reduced to an electrostatic one. The only force relevant to their dynamics arises from the in-plane component of the (total) electrostatic field described by a scalar potential (T,z,t). The equation of motion of the electron beam is thus d\/dt = (e/m)V<£ +OJ C XV where v is the velocity field and c £. If we suppose a time variation = (r,z)exp/a^, we can combine the linearized equation of motion rewritten as (0 o (/) < v= (e/m)(V 2 0+o> c x V 2 <£)/(co c 2 -co 2 ) (1) with the linearized continuity equation and a Gauss integral of Poisson's equation to write the boundary con-20 30 40 Frequency (MHz)FIG. 1. Part of the longitudinal spectrum of electrons of density « 0 =1.7xl0 7 cm" 2 obtained at T = 60 mK. The upper-plate, guard-ring, and corresponding electron-surface potentials are respectively -3, -7, and -3 V. The solid and dashed arrows of (a) indicate the calculated mode frequencies with and without the density profile correction. For this density the solid-phase vibration frequency of ...

The surface tension of liquid 4He is determined from the frequencies of micron wavelength capillary waves. The extrapolated zero temperature value, G = 375 + 3 #Jm -2, is in agreement with the pioneering static capillary rise determination but 6Uo higher than the more recent surface tension gravity wave measurements. Flow in the meniscus in this latter experiment is shown to mimic a surface tension correction to the dispersion relation there used which is of the same sign and magnitude as the discrepancy.

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The shear modulus of two-dimensional electrons at the surface of liquid helium is deduced from measurements of the finite frequency of the k-+0 limit of the coupled electronsubstrate transverse sound mode. Its behavior on the approach to melting is compatible with the Kosterlitz-Thouless model for two-dimensional melting.

Shear waves are shown to propagate in the two-dimensional electron solid on liquid helium. Their velocity and damping are measured and used to deduce the shear components of the viscoelastic tensor up to melting. A linear temperature variation at low temperatures contrasts with the premelting region, where the sharper renormalization of the elastic component is accompanied by a very rapid increase in the viscous component.PACS 63.20.Dj, 64.70.Dv A material is solid if it returns to its original shape upon removal of an applied shear stress. At finite frequency this property combines with inertia to give rise to shear waves. Shear waves are thus a fundamental attribute of a solid, but no experimental demonstration has been given that they propagate in a two-dimensional (2D) system. 1 The experiment described here on a classical 2D electron solid shows explicitly that shear (transverse phonons) does propagate and the manner in which it ceases to do so as melting is approached.These observations are presented quantitatively in terms of the generalized shear modulus for an isotropic solid, fji(k, co) = /*' +//x" = \i! +i